Molecular Spectra and Molecular Structure - Lec 6

Page 1 Lecture #6: Alkali and Many e– Atomic Spectra What am I doing spending all of this time on ATOMS? One of the goals of spectroscopy is to recover what is unique about the system by using what is universal and general as a road map. Form and flavor of electronic structure models. * patterns for assignment of spectra * predictions of unobserved states and properties * ways to estimate size and shape of orbitals * quantitative reconstruction of Vl(r) from spectrum * zero-order pictures for describing dynamics. Ψ (Q, t ) = ∑ a i (t)ψ0i (Q ) % i ψi ↔ ∑α i ψ0i time evolving non-eigenstate expressed in terms of basis states, ψoi eigenstate expressed in terms of basis states, ψoi Construct interesting Ψ(Q, 0) and predict its time evolution. How do we know n, l and Z for 1e– spectrum? pattern redundancy Alkalis * convergence ℜZ2/n2 * fine structure Z4/n3 (l + 1)(l + 1/2)l * hyperfine? * selection rules for electric dipole transition * exactly repeated intervals in two series SCF → mathematical definition of 1e– orbitals Scaling generalized ⎫ interpretive ⎪ eff ⎪ intuitive Z → Z nl ⎬ n → n* = n − δ l ⎪ diagnostic ⎪⎭ systematic probe core, valence, Rydberg core-penetrating, non-penetrating Ontogeny recapitulates phylogeny [Mulliken] What happens when you throw an e– at a closed shell ion? - intuition Page 2 - Quantum mechanical wavepacket calculation. VA ( r ) = − Z eff (r )e 2 r + = 2 A(A + 1) r2 l-dependent effective potential # of radial nodes ? 2s vs 2p? spacing of radial nodes? λ = h/p l(l+1) effect of r2 on Enl, on rnl? VA(r) A≠0 pl(r) = [(E–Vl(r))2µ]1/2 r Zeff(r)e2 r 1e2/r A=0 2 Ze /r 14.4Z VA=0(r)/eV = – r/Å Real curve is more attractive than Z = 1 curve Page 3 ask for these and other effects. Why? for n? for n*? EnA’s are lowered nodes are closer together (same node count?) inner part of RnA becomes more compact not a simple break 1e– atom degeneracy 3s 3p 3d r/Z scaling of ψ 2s 2p 1s outer part of Rnl same as Z = 1 at that value of E that corresponds to n* Shell model valence region K shell L shell z Zeff(r) ZCORE r Rydberg core valence (filled) low-n* orbitals are exclusively inside core valence orbital penetrates inside core → HOAO → n*0 low-l δl > 0 penetrating Rydberg orbitals non-penetrating high-l δl = 0 δl  > δp > δd > δf ≈ 0 see ZCORE = integer Page 4 Zneflf core orbitals - XPS spectrum core part of valence orbitals core part of penetrating Rydberg orbitals properties like spin-orbit and hyperfine Zeff > ZCORE and δ ≠ 0 and independent of n n* = n – δl outer part of valence orbitals outer part of penetrating orbitals all of non-penetrating orbitals Z = ZCORE OR ZION (integer) ontogeny recapitulates phylogeny ∑ Rydberg Series * Replicated inner lobes * n*–3/2 scale factor e– ↔ core energy exchange e– scattered off core πδl phase shift (with respect to H+ + e–) inter-channel interactions due to 1/rij with core excited states simple picture follows in order to understand δl systematics and ψn*l recapitulation. Eigenstates in this energy region asymptotic limit core levels very far apart and not following constant-Z Rydberg equa tion V(r) What do we know about this kind of potential? Are all eigenfunctions pictorially related? How do E levels tell us about form of V(r)? “Rydberg” levels close together, converging, and scalable with constant Z Page 5 Why is δl n-independent? Boundary condition: Rnl(r) → 0 at r → ∞ (so phase at r> must be just right to prevent blow-up at r → ∞) Outside “core”, Z eff nl (r) → Z CORE (integer) thus all Rnl(r) have identical asymptotic form (except for the possibility of a phase shift), because the Vl(r) → –ZCORE/r. Inside “core” — all of the extra phase accumulates because Z eff nl (r) > ZCORE. All ψn*l in given series (channel) exit core with same phase. Must splice Coulomb long range wavefunction onto core wavefunction. n=3 n=2 r> core: Vl(r) inside n = 1 region KE is enormous KE does not vary significantly with n* de Broglie λ’s (nodal structure) are independen t of n* along each series. e– exits core with same phase, independen t of n+. n=1 amplitude inside core ~ n*–3/2 period ∝ n–3 SEMI-CLASSICAL Harmonic oscillator period is T = 1 2π = ν ω −1 ⎡ E − E n −1 ⎤ Quantum Mechanical period is ⎢ n +1 → n *−3 ⎥ ⎣ ⎦ 2h ∆t inside core independent of n* ∆t –3 –3/2 T ∝ n probability inside core (amplitude inside core ∝ n ) Page 6 Quantum Defect Theory throw e– at M+ ion set of δl’s tells us about Zeff(r) from extra phase accumulated inside core region. different l’s tell us different depths of penetration — partial wave analysis. [complementary inside-core information from spin-orbit and hfs ↔ Zeff] What if e– hits a core e– and scatters it out? Perturbation of Rydberg series member n2L by a core excited state? Usually it costs too much to promote a core e–. Except for 3d104s1 (Cu, Ag, Au) (but not for a Rydberg series converging to an electronically excited state of the ion.) Doubly excited states — Rydberg series built on a core hole. e.g. Na [1s2 2s2 2p5]3s n*l 2,4 l+1 2 P CORE l l –1 are the possible states Spectrum gets very complicated at high E! Autoionization: eject e– matrix elements of 1/rij between ionization continuation and doubly excited state. Crucial differences between hydrogenic and alkali-like spectra * loss of degeneracy between n2Lj=l+1/2 and n2(L + 1)j (different shielding/core-penetration of s,p, d, f…) * loss of simple analytic f(quantum numbers) for all radial properties. Retain empirically corrected scaling relationships. Retain ability to estimate sizes. (n-independence of δ’s means that a quantitative theory exists) * possibility of core-excited states (core no longer 1S closed shell) possibility of core e–↔valence e– energy transfer “autoionization”, “superexcited” states, “resonances” * retain one doublet state for each closed-shell core plus nl electronic configuration. What happens when there is more than one e– outside of closed shells — amazing complexity!